Questions: Cauchy's Integral Formula for Derivatives

Question 1

A function f is holomorphic on a domain D. Which statement correctly describes f's differentiability?

• A. f may fail to have a second derivative at isolated points, as in real analysis
• B. f must have derivatives of all orders, and each derivative is also holomorphic
• C. f has at most finitely many derivatives, determined by its complexity
• D. f has infinitely many derivatives, but they need not be holomorphic

Answer: (B)

Holomorphicity implies infinite differentiability — a fundamental asymmetry with real analysis. In real analysis, differentiability once does not imply differentiability twice. But since Cauchy's integral formula exists for every order n (just differentiate under the integral sign n times), f^(n)(z₀) exists for all n, and each derivative is again holomorphic. Option D is tempting but wrong: each derivative is also holomorphic, not merely differentiable.

Question 2

You want to compute the n-th derivative of a holomorphic function f at z₀ using Cauchy's formula. Compared to the formula for f(z₀) itself, the formula for f^(n)(z₀):

• A. Replaces the denominator (z − z₀) with (z − z₀)^n
• B. Divides by n! to normalize for the repeated differentiation
• C. Multiplies by n! and replaces (z − z₀) with (z − z₀)^(n+1) in the denominator
• D. Is identical — you differentiate the result after evaluating the contour integral

Answer: (C)

Differentiating 1/(z − z₀) with respect to z₀ n times gives n!/(z − z₀)^(n+1) by the power rule. This factor of n! appears in the numerator and the denominator gains one extra power of (z − z₀) per differentiation. The full formula is f^(n)(z₀) = (n!/2πi) ∮ f(z)/(z − z₀)^(n+1) dz. Option A (wrong exponent) and option B (divides rather than multiplies) are the most common errors.

Question 3

In real analysis, differentiability once implies differentiability infinitely many times, just as in complex analysis.

• T. True
• F. False

Answer: False

This is one of the deepest asymmetries between real and complex analysis. In real analysis, a function can be differentiable exactly once (or any finite number of times) without being differentiable a second time. Complex differentiability (holomorphicity) is far more restrictive: the Cauchy integral formula allows recovery of every derivative from boundary values, so a holomorphic function is automatically infinitely differentiable. The rigidity of holomorphic functions — infinite differentiability, equality with their Taylor series — has no real-analysis counterpart.

Question 4

Cauchy's integral formula for the n-th derivative can be derived by differentiating the original integral formula for f(z₀) with respect to z₀ under the integral sign.

• T. True
• F. False

Answer: True

The derivation is exactly this: start from f(z₀) = (1/2πi) ∮ f(z)/(z − z₀) dz and differentiate the integrand with respect to z₀. Each differentiation applies the power rule to 1/(z − z₀), producing an extra power in the denominator and accumulating a factorial in the numerator. After n differentiations, d^n/dz₀^n[1/(z − z₀)] = n!/(z − z₀)^(n+1), giving the formula directly.

Question 5

Why does holomorphicity in complex analysis imply infinite differentiability, while differentiability in real analysis does not?

Model answer: In complex analysis, Cauchy's integral formula expresses f(z₀) as a contour integral over the boundary. Differentiating this formula with respect to z₀ under the integral sign produces a valid formula for f'(z₀), and there is no obstruction to repeating this process for any n — so all derivatives exist. In real analysis, there is no analogous integral representation that forces derivatives to exist beyond a given order; a function can have a derivative at every point without that derivative being itself differentiable.

The key is that the integral formula gives f^(n)(z₀) for every n without requiring additional assumptions. Holomorphicity is a single condition that turns out to entail the entire tower of derivatives. In real analysis, each level of differentiability is an independent condition — you must separately assume f', f'', etc. exist. This is why complex analysis has such a richer theory: one mild-seeming condition implies an enormous amount of structure.